2020/12/02 by Silvano Pitassi, Pitassi, Silvano, Francesco Trevisan +3
Computer Science · Engineering · #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #FOS: Mathematics #Numerical Analysis (math.NA) #Optical measurement and interference techniques
paper · pdf · doi:10.48550/arxiv.2012.01094
openalex publication_date 2020/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The geometric reinterpretation of the Finite Element Method (FEM) shows that\nRaviart Thomas and Nedelec mass matrices map from degrees of freedoms (DoFs)\nattached to geometric elements of a tetrahedral grid to DoFs attached to the\nbarycentric dual grid. The algebraic inverses of the mass matrices map DoFs\nattached to the barycentric dual grid back to DoFs attached to the\ncorresponding primal tetrahedral grid, but they are of limited practical use\nsince they are dense.\n In this paper we present a new geometric construction of sparse inverse mass\nmatrices for arbitrary tetrahedral grids and possibly anisotropic materials,\ndebunking the conventional wisdom that the barycentric dual grid prohibits a\nsparse representation for inverse mass matrices. In particular, we provide a\nunified framework for the construction of both edge and face mass matrices and\ntheir sparse inverses. Such a unifying principle relies on novel geometric\nreconstruction formulas, from which, according to a well established design\nstrategy, local mass matrices are constructed as the sum of a consistent and a\nstabilization term. A major difference with the approaches proposed so far is\nthat the consistent term is defined geometrically and explicitly, that is,\nwithout the necessity of computing the inverses of local matrices. This\nprovides a sensible speedup and an easier implementation. We use these new\nsparse inverse mass matrices to discretize a three dimensional Poisson problem,\nproviding the comparison between the results obtained by various formulations\non a benchmark problem with analytical solution.\n